Wednesday

July 23, 2014

July 23, 2014

Posted by **anisah** on Sunday, April 22, 2012 at 3:44am.

a)the position of the ball at any time.

b)the position of the bal at t=pai/12 sec

c)the circular frequency, natural frequency and period for this system

- differential equations -
**MathMate**, Sunday, April 22, 2012 at 9:41amThis is a "spring-mass" mass problem.

m=128 lb

L=2 ft

g=32.2 ft/sec² (accel. due to gravity)

k=mg/L=128*32.2/2=2060.8

Let y=displacement (downwards) in the same direction as g, then the differential equation that governs the motion is:

acceleration = -ky, or

y"+ky=0 where y"=d²y/dt²

The solution of the auxiliary equation is

(m^2+k)=0, or m=±i√k

and the solution to the equation is

y=C1 cos(√k t) + C2 sin(√k t)

The initial conditions are:

y(0)=-0.5, y'(0)=0.

Since

y(0)=-0.5=C1 cos(0) + C2 sin(0)

C2=0, and C1=-0.5

y'(t)=-C2√k sin(√k t) + C2 √k cos(√k t)

From

y'(0)=0=-C2√k sin(0) + C2 √k cos(0)

we conclude

C2=0 since sin(0)=0, √k≠0 and cos(0)≠0.

So the equation of motion is:

y=-(1/2)cos(√ t)

Answers to (b) and (c) can be deduced from the equation of motion.

**Related Questions**

math - a steel ball weighing 128 lb is suspended from a spring, whereupon the ...

physics - A ball of mass 540 g hangs from a spring whose stiffness is 155 ...

College Physics - A ball of mass 620 g hangs from a spring whose stiffness is ...

Physics - A spring with a spring constant of 223.9 N/m is compressed by 0.220 m...

PHYSICS - a 0.1 kg mass is suspended at rest from a spring near the Earth's ...

physics - a 0.1 kg mass is suspended at rest from a spring near the Earth's ...

physics - If a 0.1 kg mass is suspended at rest from a spring near the Earth's ...

physics - The spring in the muzzle of a child's spring gun has a spring constant...

Physics - Eqilibrium position of a spring-ball - A 275g ball is resting on top ...

Physics - Spring balls - A 275g ball is resting on top of the spring that is ...